STUDY OF DIFFERENT TYPES OF CONNECTION IN VARIOUS DIFFERENTIABLE MANIFOLD

The present work investigates numerous types of affine connections defined on differentiable manifolds, with a focus on Lorentzian para-Sasakian (LP-Sasakian) manifolds with non-Levi-Civita structures. We construct explicit equations for the related torsion, curvature, Ricci tensor, and scalar curvature by adding a semi-symmetric non-metric connection that is consistent with the underlying LP-Sasakian structure, illuminating their differences from traditional Riemannian equivalents. The paper rigorously describes generalized pseudo Ricci symmetric, generalized Ricci-recurrent, semi-pseudo symmetric, and semi-pseudo Ricci symmetric manifolds using covariant differential constraints on curvature and Ricci tensors. Several structural theorems are established utilizing tensorial identities, Bianchi-type relations, and contraction techniques, revealing that generalized Ricci-recurrent LP-Sasakian manifolds admitting Codazzi or cyclic type Ricci tensors necessarily reduce to Einstein manifolds under the considered connection.Furthermore, the non-existence of semi-pseudo symmetric and semi-pseudo Ricci symmetric LP-Sasakian manifolds (for dimension ????>3) permitting a semi-symmetric non-metric connection is convincingly shown, underlining inherent geometric impediments. The provided conclusions not only unify several ideas of curvature symmetry within a generalized affine framework, but they also provide a better understanding of how non-metricity and torsion impact the global geometric behavior of differentiable manifolds. This study adds to the larger theory of non-Riemannian geometry and establishes a foundation for future research in extended geometric structures and mathematical physics.